Bridging the Gap Between Stable Marriage and Stable Roommates: A Parameterized Algorithm for Optimal Stable Matchings

TL;DR

This paper introduces a fixed-parameter tractable algorithm for finding optimal stable matchings in the Stable Roommates problem by leveraging a new structural distance measure from Stable Marriage instances.

Contribution

It defines the minimum crossing distance and shows how to efficiently compute optimal stable matchings when this distance is small, bridging the gap between SR and SM.

Findings

Optimal stable matching can be found in time 2^{O(k)} n^{O(1)} for instances with minimum crossing distance k.

When the minimum crossing distance is zero, the SR instance is equivalent to an SM instance and can be solved efficiently.

The problem is fixed-parameter tractable with respect to the minimum crossing distance.

Abstract

In the Stable Roommates Problem (SR), a set of $2n$ agents rank one another in a linear order. The goal is to find a matching that is stable: one that has no pair of agents who mutually prefer each other over their assigned partners. We consider the problem of finding an optimal stable matching. Agents associate weights with each of their potential partners, and the goal is to find a stable matching that minimizes the sum of the associated weights. Efficient algorithms exist for finding an optimal stable matching in the Stable Marriage Problem (SM), but the problem is NP-hard for general SR instances. In this paper, we define a notion of structural distance between SR instances and SM instances, which we call the minimum crossing distance. When an SR instance has minimum crossing distance $0$, the instance is structurally equivalent to an SM instance, and this structure can be exploited to find an optimal stable matching efficiently. More generally, we show that when an SR instance has minimum crossing distance $k$, an optimal stable matching can be computed in time $2^{O(k)} n^{O(1)}$. Thus, the optimal stable matching problem is fixed-parameter tractable (FPT) with respect to minimum crossing distance.